Language model watermarking enables reliable detection of model-generated text, which has many applications in the responsible deployment of language models. Existing watermarking strategies operate by altering the decoder of an existing language model, and the ability for a language model to directly learn to generate the watermark would have significant implications for the real-world deployment of watermarks. First, learned watermarks could be used to build open models that naturally generate watermarked text, allowing for open models to benefit from watermarking. Second, if watermarking is used to determine the provenance of generated text, an adversary can damage the reputation of a victim model by spoofing its watermark and generating harmful watermarked text. To investigate the learnability of watermarks, we propose watermark distillation, which trains a student model to behave like a teacher model that uses decoding-based watermarking. We test our approach on three distinct decoding-based watermarking strategies, finding that models can learn to generate watermarked text with high detectability. We also find limitations to learnability, including the loss of watermarking capabilities under fine-tuning on normal text and high sample complexity when learning low-distortion watermarks.